Applications of homotopy theory to 4D geometry, number theory, and physics

同伦理论在四维几何、物理学和数论的最新研究中起着重要的作用：它是一种技术上强大的思想，揭示了表面上看似无关的主题之间的深层联系。马德森、蒂尔曼、韦斯、科恩和其他人最近关于黎曼曲面模空间的稳定上同调的工作，是基于弦物理学中某些协边范畴的研究。 这些结构具有四维类似物（在物理学中具有类似的意义），它们与Hatcher，Waldhausen和其他人在80年代和90年代发展的伪同位素理论有着惊人的联系。该理论涉及代数K理论的整数在一个非平凡的方式;和后一个主题现在被视为发挥了重要作用，在一个显然无关的循环的想法连接绝对伽罗瓦群的有理数领域的理论动机在代数几何（通过工作德利涅，贡恰罗夫，孔采维奇，和其他人）。这些想法反过来又涉及到工作的康纳斯，克莱默，和其他人对重新解释经典理论的重整化物理学。这一建议的中心概念之一是希望在微分拓扑中找到代数几何学家的混合泰特动机的模拟，这将与整数的代数K-理论有关，因为这一主题是瓦尔德豪森的代数K-理论的空间。在较少的技术术语：同伦理论提供了一个机构的技术研究数学对象和他们的变形在平等的基础上;事实上，它同样乐于研究形变的形变，等等，这个提议最终涉及两套源于物理学的思想，一套来自现代弦理论，另一套来自更经典的重整化理论。前一套思想与微分拓扑有着深刻的联系，而后者则与代数几何和数论的最新发展有关。从表面上看，微分拓扑学和算术代数几何学相距甚远，但它们通过整数的代数K-理论联系在一起，这是同伦理论的一部分。[因为这个主题如此系统地处理一种理论到另一种理论的变形，它提供了一套非常方便的工具来联系这些不同的主题。这个建议提出了一种澄清这些联系的方法，基于关于空间而不是数字的K理论的想法，这是由Waldhausen首先提出的，并且最近由算术几何的工作者如Soul 'e进一步货币化。该计划将以一种概念上令人满意的方式汇集数学和物理学的这些重要的最新发展。

DMS-0406461 Jack MoravaHomotopy theory plays an important role in recent work in four-dimensional geometry, physics, and number theory: it is a technically powerful ideology, which exposes deep connections among topics which may on the surface seem unrelated. Recent work of Madsen, Tillmann, Weiss, Cohen, and others on the stable cohomology of the moduli space of Riemann surfaces is based on the study of certain cobordism categories suggested by ideas from string physics. These constructions have four-dimensional analogs (of similar significance in physics), which display surprising connections to the pseudoisotopy theory developed in the 80's and 90's by Hatcher, Waldhausen, and others. That theory involves the algebraic K-theory of the integers in a nontrivial way; and the latter subject is now seen to play an important role in an apparently unrelated circle of ideas connecting the absolute Galois group of the rational number field to the theory of motives in algebraic geometry (through work of Deligne, Goncharov, Kontsevich, and others). These ideas are in turn involved in work of Connes, Kreimer, and others on a reinterpretation of the classical theory of renormalization in physics. One of the central notions of this proposal is the hope of finding in differential topology an analog of the algebraic geometers' mixed Tate motives, which would be related to the algebraic K-theory of the integers as that subject is to Waldhausen's algebraic K-theory of spaces.In less technical terms: homotopy theory provides a body of techniques for studying mathematical objects and their deformations on an equal footing; indeed, it is equally happy studying deformations of deformations, and so on. This proposal is concerned ultimately with two sets of ideas with roots in physics, one coming from modern string theory, the other from the more classical theory of renormalization. The former set of ideas has deep connections with differential topology, and the latter is related to recent developments in algebraic geometry and number theory. On the surface, differential topology and arithmetic algebraic geometry are far apart, but they are linked through the algebraic K-theory of the integers, which is at base a part of homotopy theory. [Because that subject deals so systematically with deformations of one theory into another, it provides a very convenient set of tools for relating such disparate subjects.] This proposal suggests a way of clarifying these linkages, based on ideas about the K-theory of spaces rather than numbers, which were first proposed by Waldhausen, and which have lately given further currency by workers in arithmetic geometry such as Soul\'e. This program would bring together these important recent developments in mathematics and physics in a conceptually satisfying way.